The Inverse Laplace Transform

Part 1. Relation of Laplace and Fourier Transforms

1. Confirm the Laplace transform of the test function f.

f[t_] :=  Sin[2*t]*Exp[-t] + t^2 - 3;
LaplaceTransform[f[t], t, s]

Copy the output below as the definition of F.

F[s_] :=  ?;

2. For given values of a and t, plot

Re[ F(a + i ω) (e^((a  +  i ω) t)) ].

a  =  1;
t  =  1;
p[omega_] :=  
        Re[F[a + I*omega]*Exp[(a + I*omega)*t]];
Plot[p[omega], {omega, -10, 10}, 
        PlotStyle -> RGBColor[1, 0, 0]];

3. Compare the given integral to f(t) for b = 10, 20, and 100. Repeat for a = 1.15 and t = 0.9.

b  =  10;
1/(2*Pi)*
        Integrate[
          F[a + omega*I]*Exp[(a + omega*I)*t], {omega, -b, 
            b}];
N[%, 8]

N[f[t]]

Part 2. An Application of the Residue Theorem

1. Write your explanation below:

2. Write your explanation below:

3. Find the singularities and calculate the residues for

Clear[t];
H[z_] :=  F[z]*Exp[z*t];

4. Define the parametric representations, and draw the curves (Subscript[C, 1])-(Subscript[C, 4]) that make up the curve C.

a  =  1;
b  =  3;
c  =  2;
Z1[tau_] :=  a - b*(1 - 2*tau)*I;
Z2[tau_] :=  a*(1 - tau) - tau*c + b*I;
Z3[tau_] :=  -c + b*(1 - 2*tau)*I;
Z4[tau_] :=  a*tau - c*(1 - tau) - b*I;

graph1  =  
        ParametricPlot[{Re[Z1[t]], Im[Z1[t]]}, {t, 0, 1}, 
          PlotStyle -> RGBColor[1, 0, 0], 
          DisplayFunction -> Identity];
graph2  =  
        ParametricPlot[{Re[Z2[t]], Im[Z2[t]]}, {t, 0, 1}, 
          PlotStyle -> RGBColor[0, 0, 1], 
          DisplayFunction -> Identity];
graph3  =  
        ParametricPlot[{Re[Z3[t]], Im[Z3[t]]}, {t, 0, 1}, 
          PlotStyle -> RGBColor[0, 1, 0], 
          DisplayFunction -> Identity];
graph4  =  
        ParametricPlot[{Re[Z4[t]], Im[Z4[t]]}, {t, 0, 1}, 
          PlotStyle -> RGBColor[1, 1, 0], 
          DisplayFunction -> Identity];
Show[graph1, graph2, graph3, graph4, 
        DisplayFunction -> $DisplayFunction];

4. Put your Residue Theorem calculations in the input cell below.

Compare your result to f[t].

f[t]

5. Your explanation:

6. Examine the size of the integrals for b = n and c = n with n = 10, 20, and 100. Record your results.

a  =  1;
t  =  1;
Clear[n];
b  =  n; 
c  =  n;

n  =  10;

Integral over (Subscript[C, 2])

Abs[
      NIntegrate[H[Z2[tau]]*(-a - c), {tau, 0, 1}]]

Integral over (Subscript[C, 3])

Abs[
      NIntegrate[H[Z3[tau]]*(-2*b*I), {tau, 0, 1}]]

Integral over (Subscript[C, 4])

Abs[
      NIntegrate[H[Z4[tau]]*(a + c), {tau, 0, 1}]]

Part 3. The Inversion Theorem

Put your calculations in the input cell below.  Leave each of the three sets of calculations in the worksheet.